Whisk egg yolks until thin, then while whisking, add rum, sugar and spices. Whip whites until they are stiff peaks. Fold whites into yolk mixture until it is the consistency of a batter. Reserve this batter. For each individual serving, in a 6 oz heatproof mug, take 1 Tablespoon of the reserved batter, 2 ounces of the brandy, and stir to combine. Add the hot water, and stir. Grate nutmeg over the top to garnish.
So.... this is a mess. The original (1862) recipe calls for 5 pounds of sugar to 12 eggs, which is insane -- you'd be looking at almost a cup of sugar dissolved into 1 egg yolk. Modern interpretations reduce the sugar (as above), may change out some of the brandy for rum, and some or all of the water for hot milk. The yield is a guess, but it feels about right. Some recipes call for up to 2 ounces of the batter per mug.
This variation is from looking at the original recipe and the Cocktail Virgin--Slut website, which references Cocktail Chronicles.
How to Mix Drinks or The Bon Vivant's Companion